Blackboard Bold i

If you're free to use LuaLaTeX and the unicode-math package and its \setmathfont macro, there are quite a few math fonts to choose from that provide a "double-struck" lowercase-i character.

Hopefully, one of the following eight choices will appeal to you. :-)

\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Stix Two Text} % choose the text font...

% Setting up eight [8!] math fonts
\setmathfont{Latin Modern Math}[version=lm]
\setmathfont{Cambria Math}[version=cambria]
\setmathfont{Asana Math}[version=asana]
\setmathfont{Stix Two Math}[version=stix2]
\setmathfont{XITS Math}[version=xits]
\setmathfont{TeX Gyre Termes Math}[version=termes]       
\setmathfont{TeX Gyre Pagella Math}[version=pagella]
\setmathfont{TeX Gyre DejaVu Math}[version=dejavu]

\newcommand\blurb{$\mathbb{123}\quad\mathbb{hij}\quad e^{\phi_i\mathbb{i}}$}
\begin{document}
\begin{tabular}{ll}
Latin Modern & \mathversion{lm}      \blurb\\
Cambria      & \mathversion{cambria} \blurb\\
Asana        & \mathversion{asana}   \blurb\\[1ex]
Stix Two     & \mathversion{stix2}   \blurb\\
XITS         & \mathversion{xits}    \blurb\\[1ex]
Termes       & \mathversion{termes}  \blurb\\
Pagella      & \mathversion{pagella} \blurb\\
DejaVu       & \mathversion{dejavu}  \blurb
\end{tabular}
\end{document}

Here, I use dafrick's answer at Double-struck zero and one to use the boondox-ds versions of bb fonts, designated here as \mymathbb{}. I used j in one location so that you can see it is not pixelated.

The fonts are installed via the boondox-dx package, which is not invoked below, so as not to overwrite the native \mathbb implementation otherwise available through amssymb.

\documentclass{article}
\usepackage{amsmath}
\DeclareMathAlphabet{\mymathbb}{U}{BOONDOX-ds}{m}{n}
\begin{document}
    \[
      x_i(t) = \mathop{\mathrm{Re}}(A_i e^{8000\pi\mymathbb{j}t}c^{\phi_i\mymathbb{i}}) \text{for $i = 1, 2, 3$}
    \]
\end{document}

---

FOLLOW UP

In comments, the OP asks for the \mathbb{i} from TG Pagella Math (OpenType format), but for use in pdflatex, which is not available. Since it is only a single glyph that is being requested, here is a kludge to obtain it:

Create the document TGbbi.tex as follows, and compile in Xelatex:

\documentclass{standalone}
\usepackage{unicode-math}
\setmathfont{TeX Gyre Pagella Math}
\begin{document}
$\mathbb{i}$
\end{document}

It creates the output TGbbi.pdf containing only the Pagella version of \mathbb{i}. Now, reverting back to pdflatex, we are going to call upon that graphic for use in a macro named \bbi defined as

\newcommand\bbi{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\mathbb{i}$}\includegraphics[height=\ht0]{TGbbi}}}

This macro requires the graphicx package (to import the graphic) and the scalerel package (to auto-scale it to the proper math size, taken as the vertical height of the regular \mathbb{i}). Thus, the implementation (showing use in several different math sizes) is

\documentclass{article}
\usepackage{amssymb,amsmath,scalerel,graphicx}
\newcommand\bbi{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\mathbb{i}$}\includegraphics[height=\ht0]{TGbbi}}}
\begin{document}
    \[
      x_i(t) = \mathop{\bbi\mathrm{Re}}(A_i e^{8000\pi\bbi t}c^{\phi_i\bbi}) \text{for $i = 1, 2, 3$}
    \]
\end{document}

Zoom:

The package cmathbb offers Blackboard Bold characters compatible with the Computer Modern.

It is the only package that I found to provide good results for \mathbb{i} for pdflatex.