ruby:2.5.0-alpine3.7 の Docker image に sqlite3 をインストールしようとしたときのメモです。
このときは、sqlite3 の 1.4.2 が最新でした。

make, g++, sqlite-dev を入れると install できました

/ # apk update
fetch http://dl-cdn.alpinelinux.org/alpine/v3.7/main/x86_64/APKINDEX.tar.gz
fetch http://dl-cdn.alpinelinux.org/alpine/v3.7/community/x86_64/APKINDEX.tar.gz
v3.7.3-180-g5372bc29f3 [http://dl-cdn.alpinelinux.org/alpine/v3.7/main]
v3.7.3-183-gcc9ad2b48d [http://dl-cdn.alpinelinux.org/alpine/v3.7/community]
OK: 9066 distinct packages available
/ # apk add make g++ sqlite-dev
(1/19) Upgrading musl (1.1.18-r3 -> 1.1.18-r4)
(2/19) Installing libgcc (6.4.0-r5)
(3/19) Installing libstdc++ (6.4.0-r5)
(4/19) Installing binutils-libs (2.30-r2)
(5/19) Installing binutils (2.30-r2)
(6/19) Installing gmp (6.1.2-r1)
(7/19) Installing isl (0.18-r0)
(8/19) Installing libgomp (6.4.0-r5)
(9/19) Installing libatomic (6.4.0-r5)
(10/19) Installing mpfr3 (3.1.5-r1)
(11/19) Installing mpc1 (1.0.3-r1)
(12/19) Installing gcc (6.4.0-r5)
(13/19) Installing musl-dev (1.1.18-r4)
(14/19) Installing libc-dev (0.7.1-r0)
(15/19) Installing g++ (6.4.0-r5)
(16/19) Upgrading musl-utils (1.1.18-r3 -> 1.1.18-r4)
(17/19) Installing make (4.2.1-r0)
(18/19) Installing sqlite-libs (3.25.3-r2)
(19/19) Installing sqlite-dev (3.25.3-r2)
Executing busybox-1.27.2-r7.trigger
OK: 187 MiB in 47 packages
/ # gem install sqlite3
Fetching: sqlite3-1.4.2.gem (100%)
Building native extensions. This could take a while...
Successfully installed sqlite3-1.4.2
1 gem installed
/ #

それぞれ、足りない時のエラー

他の OS でも似たようなエラーが出る可能性があると思うので記載しておきます。

make がない

/ # gem install sqlite3
Fetching: sqlite3-1.4.2.gem (100%)
Building native extensions. This could take a while...
ERROR:  Error installing sqlite3:
	ERROR: Failed to build gem native extension.

    current directory: /usr/local/bundle/gems/sqlite3-1.4.2/ext/sqlite3
/usr/local/bin/ruby -r ./siteconf20200515-13-8rgmaw.rb extconf.rb
checking for sqlite3.h... yes
checking for pthread_create() in -lpthread... yes
checking for -ldl... yes
checking for sqlite3_libversion_number() in -lsqlite3... yes
checking for rb_proc_arity()... yes
checking for rb_integer_pack()... yes
checking for sqlite3_initialize()... yes
checking for sqlite3_backup_init()... yes
checking for sqlite3_column_database_name()... yes
checking for sqlite3_enable_load_extension()... yes
checking for sqlite3_load_extension()... yes
checking for sqlite3_open_v2()... yes
checking for sqlite3_prepare_v2()... yes
checking for sqlite3_int64 in sqlite3.h... yes
checking for sqlite3_uint64 in sqlite3.h... yes
creating Makefile

current directory: /usr/local/bundle/gems/sqlite3-1.4.2/ext/sqlite3
make "DESTDIR=" clean
sh: make: not found

current directory: /usr/local/bundle/gems/sqlite3-1.4.2/ext/sqlite3
make "DESTDIR="
sh: make: not found

make failed, exit code 127

g++ がない

/ # gem install sqlite3
Fetching: sqlite3-1.4.2.gem (100%)
Building native extensions. This could take a while...
ERROR:  Error installing sqlite3:
	ERROR: Failed to build gem native extension.

    current directory: /usr/local/bundle/gems/sqlite3-1.4.2/ext/sqlite3
/usr/local/bin/ruby -r ./siteconf20200515-10-llmpwx.rb extconf.rb
*** extconf.rb failed ***
Could not create Makefile due to some reason, probably lack of necessary
libraries and/or headers.  Check the mkmf.log file for more details.  You may
need configuration options.

Provided configuration options:
	--with-opt-dir
	--without-opt-dir
	--with-opt-include
	--without-opt-include=${opt-dir}/include
	--with-opt-lib
	--without-opt-lib=${opt-dir}/lib
	--with-make-prog
	--without-make-prog
	--srcdir=.
	--curdir
	--ruby=/usr/local/bin/$(RUBY_BASE_NAME)
	--with-sqlcipher
	--without-sqlcipher
	--with-sqlite3-config
	--without-sqlite3-config
	--with-pkg-config
	--without-pkg-config
/usr/local/lib/ruby/2.5.0/mkmf.rb:456:in `try_do': The compiler failed to generate an executable file. (RuntimeError)
You have to install development tools first.
	from /usr/local/lib/ruby/2.5.0/mkmf.rb:541:in `try_link0'
	from /usr/local/lib/ruby/2.5.0/mkmf.rb:559:in `try_link'
	from /usr/local/lib/ruby/2.5.0/mkmf.rb:660:in `try_ldflags'
	from /usr/local/lib/ruby/2.5.0/mkmf.rb:1820:in `pkg_config'
	from extconf.rb:35:in `<main>'

To see why this extension failed to compile, please check the mkmf.log which can be found here:

  /usr/local/bundle/extensions/x86_64-linux/2.5.0/sqlite3-1.4.2/mkmf.log

extconf failed, exit code 1

sqlite-dev がない

/ # gem install sqlite3
Fetching: sqlite3-1.4.2.gem (100%)
Building native extensions. This could take a while...
ERROR:  Error installing sqlite3:
	ERROR: Failed to build gem native extension.

    current directory: /usr/local/bundle/gems/sqlite3-1.4.2/ext/sqlite3
/usr/local/bin/ruby -r ./siteconf20200515-10-s4bcgw.rb extconf.rb
checking for sqlite3.h... no
sqlite3.h is missing. Try 'brew install sqlite3',
'yum install sqlite-devel' or 'apt-get install libsqlite3-dev'
and check your shared library search path (the
location where your sqlite3 shared library is located).
*** extconf.rb failed ***
Could not create Makefile due to some reason, probably lack of necessary
libraries and/or headers.  Check the mkmf.log file for more details.  You may
need configuration options.

Provided configuration options:
	--with-opt-dir
	--without-opt-dir
	--with-opt-include
	--without-opt-include=${opt-dir}/include
	--with-opt-lib
	--without-opt-lib=${opt-dir}/lib
	--with-make-prog
	--without-make-prog
	--srcdir=.
	--curdir
	--ruby=/usr/local/bin/$(RUBY_BASE_NAME)
	--with-sqlcipher
	--without-sqlcipher
	--with-sqlite3-config
	--without-sqlite3-config
	--with-pkg-config
	--without-pkg-config
	--with-sqlcipher
	--without-sqlcipher
	--with-sqlite3-dir
	--without-sqlite3-dir
	--with-sqlite3-include
	--without-sqlite3-include=${sqlite3-dir}/include
	--with-sqlite3-lib
	--without-sqlite3-lib=${sqlite3-dir}/lib

To see why this extension failed to compile, please check the mkmf.log which can be found here:

  /usr/local/bundle/extensions/x86_64-linux/2.5.0/sqlite3-1.4.2/mkmf.log

extconf failed, exit code 1