The following works for me, though it may be imperfect as I'm not very comfortable with RefTeX customization:
(defun my-add-to-alist (alist-var key value &optional append)
"Add an element to an association list.
If ALIST-VAR contains no element whose CAR is `equal' to KEY, then add
(KEY . VALUE) to ALIST-VAR. If APPEND is nil, add it to the beginning of
ALIST-VAR, otherwise add it to the end of ALIST-VAR.
If there is already an element in ALIST-VAR whose CAR is `equal' to KEY,
replace its CDR with VALUE."
(interactive)
(let ((cons-cell (assoc key (eval alist-var))))
(if cons-cell
(setcdr cons-cell value)
(add-to-list alist-var (cons key value) append))))
(my-add-to-alist 'reftex-ref-style-alist
"AMSMath" '("amsmath" (("\\eqref" ?e))) t)
(require 'tex)
(TeX-add-style-hook "amsmath"
#'(lambda ()
(when (fboundp 'reftex-add-label-environments)
(add-to-list 'reftex-ref-style-default-list "AMSMath" t)
(reftex-ref-style-activate "AMSMath")
(setq reftex-label-alist '(AMSTeX)))))
Note: I assume you already have something like this before the above code:
;; Use RefTeX in LaTeX buffers
(require 'reftex)
(add-hook 'LaTeX-mode-hook 'turn-on-reftex) ; with AUCTeX's LaTeX mode
; (add-hook 'latex-mode-hook 'turn-on-reftex) ; with Emacs' latex mode
(setq reftex-plug-into-AUCTeX t)